PE000F

prime_list_nonempty_chain

Every positive-length first-prime list exposes its actual final index and minimal-successor chain.

Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Every successor is the globally least prime above its predecessor. This is not a sparse Bertrand chain. The bound theorem constructs the list and both power witnesses from k≠0 alone; the separate total-list theorem includes k=0.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ k. InitialPrimeList(b,c,k) → ¬k = 0 → ∃ x. k = S x ∧ InitialPrimeChain(b,c,x)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

InitialPrimeChain(b,c,k) · 1InitialPrimeList(b,c,k) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall b c k. ((k = 0 \/ exists pen_last_index_nonempty_list. k = S pen_last_index_nonempty_list /\ ((((exists fs_h_pen_nonempty_list_chain_initial. fs_h_pen_nonempty_list_chain_initial + S (2) = S ((S (0)) * c)) /\ exists fs_q_pen_nonempty_list_chain_initial. b = fs_q_pen_nonempty_list_chain_initial * S ((S (0)) * c) + (2))) /\ forall pen_index_nonempty_list_chain. (exists pc_lt_pen_nonempty_list_chain_bound. pc_lt_pen_nonempty_list_chain_bound + S (pen_index_nonempty_list_chain) = (pen_last_index_nonempty_list)) -> exists pen_previous_nonempty_list_chain pen_following_nonempty_list_chain. (((exists fs_h_pen_nonempty_list_chain_previous. fs_h_pen_nonempty_list_chain_previous + S (pen_previous_nonempty_list_chain) = S ((S (pen_index_nonempty_list_chain)) * c)) /\ exists fs_q_pen_nonempty_list_chain_previous. b = fs_q_pen_nonempty_list_chain_previous * S ((S (pen_index_nonempty_list_chain)) * c) + (pen_previous_nonempty_list_chain))) /\ ((((exists fs_h_pen_nonempty_list_chain_following. fs_h_pen_nonempty_list_chain_following + S (pen_following_nonempty_list_chain) = S ((S (S pen_index_nonempty_list_chain)) * c)) /\ exists fs_q_pen_nonempty_list_chain_following. b = fs_q_pen_nonempty_list_chain_following * S ((S (S pen_index_nonempty_list_chain)) * c) + (pen_following_nonempty_list_chain))) /\ (((~(pen_following_nonempty_list_chain = 1) /\ forall bpr_left_pc_pen_nonempty_list_chain_next_prime bpr_right_pc_pen_nonempty_list_chain_next_prime. pen_following_nonempty_list_chain = bpr_left_pc_pen_nonempty_list_chain_next_prime * bpr_right_pc_pen_nonempty_list_chain_next_prime -> bpr_left_pc_pen_nonempty_list_chain_next_prime = 1 \/ bpr_right_pc_pen_nonempty_list_chain_next_prime = 1)) /\ ((exists pc_lt_pen_nonempty_list_chain_next_greater. pc_lt_pen_nonempty_list_chain_next_greater + S (pen_previous_nonempty_list_chain) = (pen_following_nonempty_list_chain)) /\ forall pen_comparison_nonempty_list_chain_next. ((~(pen_comparison_nonempty_list_chain_next = 1) /\ forall bpr_left_pc_pen_nonempty_list_chain_next_comparison bpr_right_pc_pen_nonempty_list_chain_next_comparison. pen_comparison_nonempty_list_chain_next = bpr_left_pc_pen_nonempty_list_chain_next_comparison * bpr_right_pc_pen_nonempty_list_chain_next_comparison -> bpr_left_pc_pen_nonempty_list_chain_next_comparison = 1 \/ bpr_right_pc_pen_nonempty_list_chain_next_comparison = 1)) -> (exists pc_lt_pen_nonempty_list_chain_next_above. pc_lt_pen_nonempty_list_chain_next_above + S (pen_previous_nonempty_list_chain) = (pen_comparison_nonempty_list_chain_next)) -> (exists pc_le_pen_nonempty_list_chain_next_minimal. pc_le_pen_nonempty_list_chain_next_minimal + (pen_following_nonempty_list_chain) = (pen_comparison_nonempty_list_chain_next)))))))) -> ~(k = 0) -> exists j. k = S j /\ ((((exists fs_h_pen_nonempty_chain_initial. fs_h_pen_nonempty_chain_initial + S (2) = S ((S (0)) * c)) /\ exists fs_q_pen_nonempty_chain_initial. b = fs_q_pen_nonempty_chain_initial * S ((S (0)) * c) + (2))) /\ forall pen_index_nonempty_chain. (exists pc_lt_pen_nonempty_chain_bound. pc_lt_pen_nonempty_chain_bound + S (pen_index_nonempty_chain) = (j)) -> exists pen_previous_nonempty_chain pen_following_nonempty_chain. (((exists fs_h_pen_nonempty_chain_previous. fs_h_pen_nonempty_chain_previous + S (pen_previous_nonempty_chain) = S ((S (pen_index_nonempty_chain)) * c)) /\ exists fs_q_pen_nonempty_chain_previous. b = fs_q_pen_nonempty_chain_previous * S ((S (pen_index_nonempty_chain)) * c) + (pen_previous_nonempty_chain))) /\ ((((exists fs_h_pen_nonempty_chain_following. fs_h_pen_nonempty_chain_following + S (pen_following_nonempty_chain) = S ((S (S pen_index_nonempty_chain)) * c)) /\ exists fs_q_pen_nonempty_chain_following. b = fs_q_pen_nonempty_chain_following * S ((S (S pen_index_nonempty_chain)) * c) + (pen_following_nonempty_chain))) /\ (((~(pen_following_nonempty_chain = 1) /\ forall bpr_left_pc_pen_nonempty_chain_next_prime bpr_right_pc_pen_nonempty_chain_next_prime. pen_following_nonempty_chain = bpr_left_pc_pen_nonempty_chain_next_prime * bpr_right_pc_pen_nonempty_chain_next_prime -> bpr_left_pc_pen_nonempty_chain_next_prime = 1 \/ bpr_right_pc_pen_nonempty_chain_next_prime = 1)) /\ ((exists pc_lt_pen_nonempty_chain_next_greater. pc_lt_pen_nonempty_chain_next_greater + S (pen_previous_nonempty_chain) = (pen_following_nonempty_chain)) /\ forall pen_comparison_nonempty_chain_next. ((~(pen_comparison_nonempty_chain_next = 1) /\ forall bpr_left_pc_pen_nonempty_chain_next_comparison bpr_right_pc_pen_nonempty_chain_next_comparison. pen_comparison_nonempty_chain_next = bpr_left_pc_pen_nonempty_chain_next_comparison * bpr_right_pc_pen_nonempty_chain_next_comparison -> bpr_left_pc_pen_nonempty_chain_next_comparison = 1 \/ bpr_right_pc_pen_nonempty_chain_next_comparison = 1)) -> (exists pc_lt_pen_nonempty_chain_next_above. pc_lt_pen_nonempty_chain_next_above + S (pen_previous_nonempty_chain) = (pen_comparison_nonempty_chain_next)) -> (exists pc_le_pen_nonempty_chain_next_minimal. pc_le_pen_nonempty_chain_next_minimal + (pen_following_nonempty_chain) = (pen_comparison_nonempty_chain_next))))))

Complete tactic proof in conservative notation

All 10 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

10 script commands · 3 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–5

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro k
  4. L4
    intro hl
  5. L5
    intro hk
02Separate the logical casesL6–7

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L6
    cases hl
  2. L7
    exfalso
03Use earlier factsL8–10

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L8
    apply hk
  2. L9
    exact hl_left
  3. L10
    exact hl_right

Library-wide reading audit

Original defined command ledger · 10 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro k
  4. 0004intro hl
  5. 0005intro hk
  6. 0006cases hl
  7. 0007exfalso
  8. 0008apply hk
  9. 0009exact hl_left
  10. 0010exact hl_right